Sign Operator for Coping with Heavy-Tailed Noise in Non-Convex Optimization: High Probability Bounds Under $(L_0, L_1)$-Smoothness
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| Formato: | Preprint |
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2025
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| _version_ | 1866913860521295872 |
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| author | Kornilov, Nikita Zmushko, Philip Semenov, Andrei Ikonnikov, Mark Gasnikov, Alexander Beznosikov, Alexander |
| author_facet | Kornilov, Nikita Zmushko, Philip Semenov, Andrei Ikonnikov, Mark Gasnikov, Alexander Beznosikov, Alexander |
| contents | In recent years, non-convex optimization problems are more often described by generalized $(L_0, L_1)$-smoothness assumption rather than standard one. Meanwhile, severely corrupted data used in these problems has increased the demand for methods capable of handling heavy-tailed noises, i.e., noises with bounded $κ$-th moment. Motivated by these real-world trends and challenges, we explore sign-based methods in this setup and demonstrate their effectiveness in comparison with other popular solutions like clipping or normalization.
In theory, we prove the first-known high probability convergence bounds under $(L_0, L_1)$-smoothness and heavy-tailed noises with mild parameter dependencies. In the case of standard smoothness, these bounds are novel for sign-based methods as well. In particular, SignSGD with batching achieves sample complexity $\tilde{O}\left(\left(\frac{ΔL_0d}{\varepsilon^2} + \frac{ΔL_1d^\frac{3}{2}}{\varepsilon}\right)\left[1 + \left(\fracσ{\varepsilon}\right)^\fracκ{κ-1}\right]\right), κ\in (1,2]$. Under the assumption of symmetric noises, SignSGD with Majority Voting can robustly work on the whole range of $κ\in (0,2]$ with complexity $\tilde{O}\left(\left(\frac{ΔL_0d}{\varepsilon^2} + \frac{ΔL_1d^\frac{3}{2}}{\varepsilon}\right)\left[\frac{1}{κ^2} + \frac{σ^2}{\varepsilon^2}\right]\right)$. We also obtain results for parameter-agnostic setups, Polyak-Lojasiewicz functions and momentum-based methods (in expectation). Our theoretical findings are supported by the superior performance of sign-based methods in training Large Language Models compared to clipping and normalization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_07923 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sign Operator for Coping with Heavy-Tailed Noise in Non-Convex Optimization: High Probability Bounds Under $(L_0, L_1)$-Smoothness Kornilov, Nikita Zmushko, Philip Semenov, Andrei Ikonnikov, Mark Gasnikov, Alexander Beznosikov, Alexander Optimization and Control Machine Learning In recent years, non-convex optimization problems are more often described by generalized $(L_0, L_1)$-smoothness assumption rather than standard one. Meanwhile, severely corrupted data used in these problems has increased the demand for methods capable of handling heavy-tailed noises, i.e., noises with bounded $κ$-th moment. Motivated by these real-world trends and challenges, we explore sign-based methods in this setup and demonstrate their effectiveness in comparison with other popular solutions like clipping or normalization. In theory, we prove the first-known high probability convergence bounds under $(L_0, L_1)$-smoothness and heavy-tailed noises with mild parameter dependencies. In the case of standard smoothness, these bounds are novel for sign-based methods as well. In particular, SignSGD with batching achieves sample complexity $\tilde{O}\left(\left(\frac{ΔL_0d}{\varepsilon^2} + \frac{ΔL_1d^\frac{3}{2}}{\varepsilon}\right)\left[1 + \left(\fracσ{\varepsilon}\right)^\fracκ{κ-1}\right]\right), κ\in (1,2]$. Under the assumption of symmetric noises, SignSGD with Majority Voting can robustly work on the whole range of $κ\in (0,2]$ with complexity $\tilde{O}\left(\left(\frac{ΔL_0d}{\varepsilon^2} + \frac{ΔL_1d^\frac{3}{2}}{\varepsilon}\right)\left[\frac{1}{κ^2} + \frac{σ^2}{\varepsilon^2}\right]\right)$. We also obtain results for parameter-agnostic setups, Polyak-Lojasiewicz functions and momentum-based methods (in expectation). Our theoretical findings are supported by the superior performance of sign-based methods in training Large Language Models compared to clipping and normalization. |
| title | Sign Operator for Coping with Heavy-Tailed Noise in Non-Convex Optimization: High Probability Bounds Under $(L_0, L_1)$-Smoothness |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2502.07923 |